Abstract

This article discusses a more general contractive condition for a class of extended (p ≥ 2) -cyclic self-mappings on the union of a finite number of subsets of a metric space which are allowed to have a finite number of successive images in the same subsets of its domain. If the space is uniformly convex and the subsets are non-empty, closed and convex, then all the iterates converge to a unique closed limiting finite sequence which contains the best proximity points of adjacent subsets and reduces to a unique fixed point if all such subsets intersect.

1. Introduction

A general contractive condition of rational type has been proposed in [1, 2] for a partially ordered metric space. Results about the existence of a fixed point and then its uniqueness under supplementary conditions are proved in those articles. The general rational contractive condition of [3] includes as particular cases several of the existing ones [1, 4–12] including Banach's principle [5] and Kannan's fixed point theorems [4, 8, 9, 11]. The general rational contractive conditions of [1, 2] are applicable only on distinct points of the considered metric spaces. In particular, the fixed point theory for Kannan's mappings is extended in [4] by the use of a non-increasing function affecting to the contractive condition and the best constant to ensure that a fixed point is also obtained. Three fixed point theorems which extended the fixed point theory for Kannan's mappings were proved in [11]. On the other hand, important attention has been paid during the last decades to the study of standard contractive and Meir-Keeler-type contractive cyclic self-mappings (see, for instance, [13–22]). More recent investigation about cyclic self-mappings is being devoted to its characterization in partially ordered spaces and to the formal extension of the contractive condition through the use of more general strictly increasing functions of the distance between adjacent subsets. In particular, the uniqueness of the best proximity points to which all the sequences of iterates converge is proven in [14] for the extension of the contractive principle for cyclic self-mappings in uniformly convex Banach spaces (then being strictly convex and reflexive [23]) if the p subsets Ai⊂X of the metric space (X, d), or the Banach space (X, || ||), where the cyclic self-mappings are defined are non-empty, convex and closed. The research in [14] is centred on the case of the cyclic self-mapping being defined on the union of two subsets of the metric space. Those results are extended in [14] for Meir-Keeler cyclic contraction maps and, in general, for the self-mapping T:⋃i∈p̄Ai→⋃i∈p̄Ai be a p(≥ 2) -cyclic self-mapping being defined on any number of subsets of the metric space with p̄:=1,2,...,p.

Other recent researches which have been performed in the field of cyclic maps are related to the introduction and discussion of the so-called cyclic representation of a set M, decomposed as the union of a set of non-empty sets as M=⋃i=1mMi, with respect to an operator f: M → M[24]. Subsequently, cyclic representations have been used in [25] to investigate operators from M to M which are cyclic φ-contractions, where φ: R0+ → R0+ is a given comparison function, M⊂X and (X, d) is a metric space. The above cyclic representation has also been used in [26] to prove the existence of a fixed point for a self-mapping defined on a complete metric space which satisfies a cyclic weak φ-contraction. In [27], a characterization of best proximity points is studied for individual and pairs of non-self-mappings S, T: A → B, where A and B are non-empty subsets of a metric space. In general, best proximity points do not fulfil the usual "best proximity" condition x = Sx = Tx under this framework. However, best proximity points are proven to jointly globally optimize the mappings from x to the distances d(x, Tx) and d(x, Sx). Furthermore, a class of cyclic φ-contractions, which contain the cyclic contraction maps as a subclass, has been proposed in [28] to investigate the convergence and existence results of best proximity points in reflexive Banach spaces completing previous related results in [14]. Also, the existence and uniqueness of best proximity points of p(≥ 2) -cyclic φ-contractive self-mappings in reflexive Banach spaces has been investigated in [29].

In this article, it is also proven that the distance between the adjacent subsets Ai , Ai+1⊂X are identical if the p(≥ 2) -cyclic self-mapping is non-expansive [16]. This article is devoted to a generalization of the contractive condition of [1] for a class of extended cyclic self-mappings on any number of non-empty convex and closed subsets Ai⊂X, i∈p̄. The combination of constants defined the contraction may be different on each of the subsets and only the product of all the constants is requested to be less than unity. On the other hand, the self-mapping can perform a number of iterations on each of the subsets before transferring its image to the next adjacent subset of the p(≥ 2) -cyclic self-mapping. The existence of a unique closed finite limiting sequence on any sequence of iterates from any initial point in the union of the subsets is proven if X is a uniformly convex Banach space and all the subsets of X are non-empty, convex and closed. Such a limiting sequence is of size q ≥ p (with the inequality being strict if there is at least one iteration with image in the same subset as its domain) where p of its elements (all of them if q = p) are best proximity points between adjacent subsets. In the case that all the subsets Ai⊂X intersect, the above limit sequence reduces to a unique fixed point allocated within the intersection of all such subsets.

2. Main results for non-cyclic self-mappings

Let (X, d) be a metric space for a metric d: X × X → R0+ with a self-mapping T: X → X which has the following contractive condition proposed and discussed in [1]:

dTx,Ty≤αdx,Txdy,Tydx,y+βdx,y,x,y≠x∈X

(2.1)

for some real constants α, β∈R0+ and α + β < 1 where R0+ = {r∈R: r ≥ 0}. A more general one involving powers of the distance is the following:

dsx,yTx,Ty≤αdσx,yx,Txdrx,yy,Tydσx,yx,y+βdtx,yx,y,x,y≠x∈X,

(2.2)

where s, σ, r, t: X × X → R+ = {r∈R: r > 0} are continuous and symmetric with respect to the order permutation of the arguments x and y. It is noted that if x = y then (2.1) has a sense only if x is a fixed point, i.e. x = y = Tx = Ty implies that (2.1) reduces to the inequality "0 ≤ 0". The following result holds:

what proves that Tnxn∈N0 is a Cauchy sequence. Such a Cauchy sequence has a limit z=limn→∞Tnx in X if (X, d) is complete from the convergence property of Cauchy sequences to points of the space X. If, in addition, T: X → X is continuous then Tz=Tlimn→∞Tnx=limn→∞Tn+1x=z so that the limit of the sequence is a fixed point. The uniqueness of the fixed point is now proven (i.e. z is not dependent on of x) by contradiction. Assume that there exists two distinct fixed points y = Ty and z = Tz in X. Then, from (2.5):

if x, Tx∈Ai provided that the following upper-bounding conditions hold:

drx,xx,Tx≤Pids(x,x)x,Tx;dtx,xx,Tx≤Qids(x,x)x,Tx

(3.5)

αi,βi,Pi,Qi≥0

Thus, the following technical result holds which does not require completeness of the metric space, uniform convexity assumption on some associated Banach space or particular properties of the non-empty subsets Ai ; ∀i∈p̄. The result will be then used to obtain the property of convergence of the sequences of iterates to best proximity points allocated in the various subsets.

Remark 3.2. It is noted that if Ai∩Aℓ=∅ and x∈Aj for some i,ℓ≠i∈p̄ and jℓ < ji then dTn+1q+ℓx,Tnq+ℓx→D as n → ∞ for all ℓ < ji . □

The following result is concerned with the proved property that distances of iterates obtained through the composed self-mapping Tq:⋃i∈p̄Ai→⋃i∈p̄Ai starting from a point x in any of the subsets, and located within two distinct of such subsets for all the iteration steps, asymptotically converge to the distance D between such subsets in uniformly convex Banach spaces, with at least one of them being convex. It is also obtained a convergence property of the iterates of the composed self-mapping Tq:⋃i∈p̄Ai→⋃i∈p̄Ai to limit points within each of the subsets.

Lemma 3.3: Let (X, || ||) be a uniformly convex Banach space endowed with the norm || || and let d: X × X →R0+ be a metric induced by such a norm || || so that (X, d) is a complete metric space. Assume that the non-empty subsets Ai of X and the extended p(≥ 2) -cyclic self-mapping T:⋃i∈p̄Ai→⋃i∈p̄Ai fulfil the constraints of Theorem 3.1 and, furthermore, one subset is closed and another one is convex and closed in each pair (Ai , Ai+1) of adjacent subsets, ∀i∈p̄. Then, the following properties hold:

; ∀x∈Ai , i ≤ m' < m < p+i, jp+i= jp , m,m′∈p̄,∀n′∈N,∀i∈p̄. Furthermore, if Ai+1is convex, then Tnq+jix→zi+1∈Ai+1 as n → ∞. Also, Tnq+∑ℓ=1mjℓx→zi+m+1=T∑ℓ=1mjℓzi∈Ai+m+1 as n → ∞ with zi+m+1≡ zi+m+1-p, Ai+m+1≡ Ai+m+1-pif m > p+1-i. Furthermore, if all the subsets Aii∈p̄ are closed and convex, then Tqnx→z∈⋂i∈p̄Ai=Tqz as n → ∞ if D = 0, that is if ⋂i∈p̄Ai≠∅, so that z∈⋂i∈p̄Ai is the unique fixed point of Tq:⋃i∈p̄Ai→⋃i∈p̄Ai in ⋂i∈p̄Ai.

(ii)

If Ai or Ai+m+1is convex then

limn→∞dTn+n′q+∑ℓ=imjℓx,Tnq+∑ℓ=imjℓx=0

(3.25)

; ∀x∈Ai,i≤m<p+i,jp+i=jp,∀n′∈N0:=N∪0,∀i∈p̄

Proof: Note from (3.6a) that

dTnq+jix,Tnqx→D∧dTn+n′q+jix,Tnqx→D⇒dTn+n′q+jix,Tnq+jix→0asn→∞

(3.26a)

; ∀x∈Ai , jp+i= jp , ∀n'∈N, ∀i∈p̄ with Tnqx∈Ai , Tnq+jix,Tn+n′q+jix∈Ai+1 with Ai+1≡ Ai+1-pif i > p-1, since (X, || ||) is a uniformly convex Banach space, d: X × X → R0+be a metric induced by the norm || ||, so that (X, d) is a complete metric space, and Ai and Ai+1are non-empty closed subsets of X and at least one of them is convex (see Lemma 3.8 of [13]). Then limn→∞dTn+n′q+jix,Tnq+jix=0. On the other hand, limn→∞dTn+n′q+∑ℓ=imjℓx,Tnq+∑ℓ=imjℓx=0 is proven by replacing (3.26a) by

dTnq+∑ℓ=imjℓx,Tnqx→D∧dTn+n′q+∑ℓ=imjℓx,Tnqx→D⇒dTn+n′q+∑ℓ=imjℓx,Tnq+∑ℓ=imjℓx→0 as n→∞

(3.26b)

with i ≤ m < p+i, ∀n'∈N. Thus, x∈Ai implies Tnqx∈Ai , Tnq+∑ℓ=imjℓx,Tn+n′q+∑ℓ=imjℓx∈Ai+m+1. The identities (3.24a) have been proven. To prove (3.24b), note from Equation 3.24a of Property (i) and the triangle inequality that the following holds:

; ∀x∈Ai , i ≤ m' < m < p+i, jp+i= jp , ∀m,m′∈p̄,∀n′∈N,∀i∈p̄. ∃limn→∞dTn+n′q+∑ℓ=imjℓx,Tnq+∑ℓ=im′jℓx=limn→∞dTn+n′q+jix,Tnqx=D. On the other hand, note that Tnq+jixn∈N is a Cauchy sequence since limn→∞n′→∞dTn+n′q+∑ℓ=imjℓx,Tnq+∑ℓ=imjℓx=0 from (3.24a) which then has a limit in X which is also in the closed and convex subset Ai+1of X. The proof of Tnq+∑ℓ=imjℓx→zi+m+1∈Ai+m+1 with Ai+m+1≠ Ai , since m < p+i, as n → ∞ follows from similar arguments since one of the subsets in each adjacent pair of subsets is convex and both of them are closed by assumption so that Tnq+∑ℓ=imjℓx=T∑ℓ=i+1mjℓTnq+jix→zi+m+1=T∑ℓ=i+1mjℓzi+1 as n → ∞; ∀x∈Ai , ∀i∈p̄. Finally, if the subsets intersect and are closed and convex then the composed self-mapping Tq:⋃i∈p̄Ai→⋃i∈p̄Ai is contractive, then continuous everywhere in its definition domain, so that it converges to a unique fixed point in the non-empty, closed and convex set ⋂i∈p̄Ai. Hence, Property (i).

To prove Property (ii), note from (3.6d) with m = i and m1 = 0 that

([d(T(n+n′)q+∑ℓ=imjℓx,Tnqx)→D]∧[d(Tnq+∑ℓ=imjℓx,Tnqx)→D])⇒(d(T(n+m)q+jix,Tnq+jix)→0) as n→∞

Assume that ⋂i∈p̄Ai≠∅. Then, the self-mapping T:⋃i∈p̄Ai→⋃i∈p̄Ai has a unique fixed point z∈⋂i∈p̄Ai. Then, any q-tuple of sequences (3.31) converges to a unique limit q-tuple (3.30) of the form ẑ:=z,....,z for any x∈⋃i∈p̄Ai and for any i∈p̄.

Proof: To keep a coherent treatment with the previous part of the manuscript and, since (X, || ||) is a Banach space with norm || ||, we can use a norm-induced metric d: X × X→R0+ which is equivalent to any other metric and then apply Theorem 3.1 to the metric space (X, d) which is complete since (X, || ||) is a Banach space. Assume the following cases:

(A)

D = 0 so that Ai∩Aj≠∅ for i,j≠i∈p̄; i.e. all the subsets have a non-empty intersection. Then, dTnq+jix,Tnqx→0, d(T(n+1)qx, Tnqx) → 0 and d(Tnq+ℓx, Tnqx) → 0; ∀x∈Ai ; ∀ℓ∈∑k=im+i-1jk:m∈p¯, ∀i∈p̄ as n → ∞ from Theorem 3.1, Equations 3.6 and 3.8, with Aj ≡ Aj-pfor 2p ≥ j > p. Thus, Tqnx(∈Ai ) → z, since {Tqnx}n∈Nis a Cauchy sequence (and also Tqn+ℓx(∈Ai+ℓ) → z) for some z∈⋂i∈p̄Ai, since ⋂i∈p̄Ai is non-empty, convex and closed from Banach contraction principle since k < 1. Since k < 1, the composed self-mapping Tq:⋃i∈p̄Ai→⋃i∈p̄Ai is contractive, and then continuous, and since (X, d) is complete, since the associated (X, || ||) is a Banach space, z∈⋂i∈p̄Ai is a unique fixed point of Tq:⋃i∈p̄Ai→⋃i∈p̄Ai. Thus, again the continuity of Tq:⋃i∈p̄Ai→⋃i∈p̄Ai and the fact that it has a unique fixed point z leads to the identities Tq (Tz) = Tq+1z = T(Tqz) = Tz = Tq (Tqz) = Tqz so that z = Tz and then z is also a fixed point of T:⋃i∈p̄Ai→⋃i∈p̄Ai. Furthermore, z is also the unique fixed point of T:⋃i∈p̄Ai→⋃i∈p̄Ai as follows by contradiction. Assume that z is not unique. Then, such that z is the unique fixed point of Tq:⋃i∈p̄Ai→⋃i∈p̄Ai and ∃y≠z∈⋂i∈p̄Ai such that y and z are both fixed points of T:⋃i∈p̄Ai→⋃i∈p̄Ai. Then, Tqy = T(Tqy) = Ty = y which contradicts that z is the unique fixed point of Tq:⋃i∈p̄Ai→⋃i∈p̄Ai. Finally, as a result of the uniqueness of the fixed point, it follows directly that any q-tuple (3.30) converges to a unique q-tuple ẑ:=z,....,z=ẑi; ∀i∈p̄ for any x∈⋃i∈p̄Ai. Hence, Property (ii).

, that is, the distance between the subsets Ai+m+1(≡ Ai+m+1-pif m > p+1-i) and Ai+m'+1(≡ Ai+m'+1-pif m > m' > p+1-i) of X equalizes that of two corresponding best proximity points, for some convergent subsequences Tmkq+∑ℓ=imjℓxi∈Ai+m+1 and Tnkq+∑ℓ=imjℓxi∈Ai+m′+1 and two best proximity points: ωi+m+1∈Ai+m+1; ωi+m'+1∈Ai+m'+1. Then, again from Lemma 3.3 and (3.34), one can get

Also, one has from Lemma 3.3, that limn→∞dTnq+∑ℓ=imjℓxi,Tn+1q+∑ℓ=imjℓxi=0 so that, by taking into account (3.35), Tnq+∑ℓ=imjℓxi→ωi+m+1∈Ai+m+1≡Ai+m+1-pifm>p+1-i as n → ∞; x∈Ai , ∀i,m≥i∈p̄ with jℓ = jℓ-pfor any ℓ > p, ∀n'∈N0, since Tnq+∑ℓ=imjℓxi∈Ai+m+1. That is, Tnq+∑ℓ=imjℓxi converges to a best proximity point of Ai+m+1; ∀xi∈Ai ; ∀i∈p̄.

Now, take a sequence Tnq+∑ℓ=imjℓ+jxi where j∈jm+1-1¯. Then, Tnq+∑ℓ=imjℓ+jxi∈Ai+m+1; ∀j∈jm+1-1¯. Assume that Tnq+∑ℓ=imjℓ+jxin∈N0 does not converge in Ai+m+1so that one can get from Theorem 3.1, Equation 3.8 and Lemma 3.3 (ii):

which is a contradiction. Then, ∃limn→∞dTn+1q+∑ℓ=imjℓ+jxi,Tnq+∑ℓ=imjℓxi=0 and Tnq+∑ℓ=imjℓ+jxin∈N0 is a Cauchy sequence with a limit in the closed and convex Ai+m+1∈X for any j∈jm+1-1¯, ∀xi∈Ai and i∈p̄. It has been proven that x^qn→ẑi; ∀i∈p̄ and the set ωi:i∈p̄ is a set of best proximity points with ωi+1=Tjiωi=Tji+qωi; ∀i∈p̄ and ωp+1= ω1. It remains to prove that the elements of the limit sequence ẑi are not dependent on the initial point xi∈Ai ; ∀i,m≥i∈p̄ to construct any sequence of iterates except, perhaps, in the order that the limiting points are allocated within such a limiting sequence. Proceed by contradiction by assuming that there are two distinct best proximity points ωi , zi∈Ai for some i∈p̄ so that:

D=dTjiωi,ωi<dTjiωi,zi=dTji+nqωi,Tnqzi

(3.37)

since ωi , zi (≠ ωi ) ∈Ai are best proximity points and Tjiωi∈Ai+1. Since that the above property holds irrespective of the integers i,m≥i∈p̄ and n∈N0, the following contradiction follows from (3.37), and Theorem 3.1, Equation 3.6a:

irrespective of i∈p̄. Therefore, the best proximity points are unique within each of the subsets. Furthermore, since T:⋃i∈p̄Ai→⋃i∈p̄Ai, the limit sequence (3.30) is unique by successive iterations from any of the best proximity points. Since there is a convergence to it from any initial point in ⋃i∈p̄Ai, any q-sequence of iterates converges to such a limit sequence, irrespective of the initial point, except for the order of the elements. □

Remark 3.5. Concerning with Theorem 3.4 (ii), note that T:⋃i∈p̄Ai→⋃i∈p̄Ai is not necessarily contractive when D = 0 although Tq:⋃i∈p̄Ai→⋃i∈p̄Ai be contractive. However, the contractive property of Tq:⋃i∈p̄Ai→⋃i∈p̄Ai leads to that of T:⋃i∈p̄Ai→⋃i∈p̄Ai possessing also a unique fixed point in ⋂i∈p̄Ai which is that of Tq:⋃i∈p̄Ai→⋃i∈p̄Ai. It is also noted that the limit sequence is unique except in the order of the elements in the sense that if a sequence of iterates x^qn Equation 3.31 converges to ẑi Equation 3.30 for any initial point x∈Ai , then for x∈Ak the limit sequence being asymptotically reached will be:

Example 3.6: Take p = 2 and subsets A1 ≡ C(a, 0, a-a0) and A2 ≡ C(-a, 0, a-a0) of R2 are circles of centre in (a, 0) and (-a, 0), respectively, and radius a-a0 with a > a0 which are defined by

A1:={(x,y)∈R2:(x−a)2+y2≤(a−a0)2};A2:={(x,y)∈R2:(x+a)2+y2≤(a−a0)2}

(3.40)

We consider the complete metric space (R2, d) with the Euclidean metric. It is clear that such a space being considered as the Banach space (R2, || ||), endowed with the Euclidean norm, is uniformly convex, then strictly convex and reflexive [23]. It is noted that D = 2a0 = dist(A1, A2) and now consider the constraint (3.1) with functions s, x, r, t: (A1∪A2) × (A1∪A2) → R+ being constant identically unity. Assume that T is some extended 2-cyclic self-mapping on A1∪A2 with j1 = 2, j2 = 1. Thus, one can get

g=x,y∈A1⇒Tg∈A1,T2g∈A2,T3g∈A1,T4g∈A1,T5g∈A2,

(3.41a)

g=x,y∈A2⇒Tg∈A1,T2g∈A1,T3g∈A2,T4g∈A1,T5g∈A1,

(3.41b)

Now, define also the family of parameterized circles A^a1:=x,y∈R2:x-a2+y2≤a12⊆A1 being with A1 and contained in it (with proper or improper set inclusion) of radius 0 < a1 ≤ a-a' for a'∈ [0, a). It is noted that A^a≡A1. Next, we define constructively a self-mapping which is an extended cyclic one and which verifies the properties. Now, consider a self-mapping T: A1∪A2 → A1∪A2 defined as follows:

It is noted that such a positive solution always exists everywhere in A1 since a≥a1≥a1-x02x02+y02. Thus, the constraint (3.1) is fulfilled by any self-mapping T: A1∪A2 → A1∪A2 with 1 > α1 ≥ 0, β1=aa01-α1>1-α1 and k1:=k̄12=aa02=β11-α12>1. It is noted that the condition (3.1) is not guaranteed to be contractive for any point of A1. It is also noted that if (x0, y0) ∈Fr(A1) then k^1x0,y0=1 with a0 = 0 so that g1 = Tg0 = g0. However, it can be noticed that g1 is not a fixed point since g2 = Tg1 ≠ g1. Next, define:

under the contractive constant k2 defined as follows subject to constraints:

k2:=β21-α2<k1-1=a0a2=1-α1β12

(3.46)

so that the composed extended cyclic self-mapping T3: A1∪A2 → A1∪A2 is subject to a the contractive condition (3.1) of contractive constant k = k1k2 < 1 with s, x, r, t: (A1∪A2) × (A1∪A2) → R+, i.e. for the Euclidean distance. Consider initial points in A2 as g0 = (x0, y0) ∈A2. Thus, (a) first apply (3.45) with the replacements (x3, y3) → (x1, y1) ∈A2 and g3 → g1(x1, y1) ∈A1; (b) then apply (3.42) with the replacement (x3, y3) → (x1, y1) and g1 →g2 = (x2, y2) ∈A1; (c) later on apply (3.44) with the replacement (x2, y2) → (x3, y3) and g2 → g3(x3, y3) ∈A2. Theorems 3.1 and 3.4 are fulfilled and there is a limiting repeated sequence of three points S*: = ((-a0, 0)), (a0, 0), (a0, 0)) which are two best proximity points, one located in A2 and another one in A1, repeated. It is noted that the repeated value of the best proximity point (a0, 0) in the limiting sequence is due to the fact that the circumference being the boundary of A1 is invariant under T, although infinitely many different cyclic self-mappings can be defined on the same two subsets A1, 2 so that a limiting sequence is reached having a common (perhaps only) the best proximity points with S*. It is also noted that if a0 ≠ 0 then (a0, 0) is not a fixed point of T: A1∪A2 → A1∪A2 since T(a0, 0) = (a0, 0) ⇒T2(a0, 0) = (-a0, 0). However, if a0 = D = 0 then the intersection of both circles is {(0, 0)} so that the limiting sequence consists of a repeated fixed point.

Declarations

Acknowledgements

The authors wish to thank the Spanish Ministry of Education for their support through Grant DPI2009-07197. They are also grateful to the Basque Government for their support through Grants IT378-10, and SAIOTEK SPE09UN12. The authors thank the referees for their useful suggestions.

Competing interests

The authors declare that they have no competing interests.

Authors' contributions

Both the authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript

Copyright

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